Question

use the given function to complete parts (a) through (e) below. f(x)=x^4 - 9x^2 a) use the leading coefficient test to determine the graphs end behavior. a. the graph of f(x) rises left and rises right. b. the graph of f(x) falls left and falls right. c. the graph of f(x) falls left and rises right. d. the graph of f(x) rises left and falls right. b) find the x - intercepts. x=-3,0,3 (type an integer or a decimal. use a comma to separate answers as needed.) at which zeros does the graph of the function cross the x - axis? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. x= (type an integer or a decimal. use a comma to separate answers as needed.) b. there are no x - intercepts at which the graph crosses the x - axis.

Explanation:

Step1: Analyze leading - coefficient and degree

The function $f(x)=x^{4}-9x^{2}$ is a polynomial of degree $n = 4$ (even) and leading coefficient $a = 1$ (positive). According to the leading - coefficient test, for a polynomial $y = a_{n}x^{n}+a_{n - 1}x^{n - 1}+\cdots+a_{0}$ with $n$ even and $a_{n}>0$, as $x\to-\infty$, $y\to+\infty$ and as $x\to+\infty$, $y\to+\infty$. So the graph of $f(x)$ rises left and rises right.

Step2: Find x - intercepts

Set $f(x)=0$, so $x^{4}-9x^{2}=0$. Factor out $x^{2}$: $x^{2}(x^{2}-9)=0$. Then factor $x^{2}-9$ further: $x^{2}(x - 3)(x + 3)=0$. Using the zero - product property, $x^{2}=0$ gives $x = 0$, $x-3=0$ gives $x = 3$, and $x + 3=0$ gives $x=-3$.

Step3: Determine where graph crosses x - axis

For a polynomial function $y = f(x)$, if the multiplicity of a zero $c$ is odd, the graph of the function crosses the $x$ - axis at $x = c$, and if the multiplicity is even, the graph touches the $x$ - axis at $x = c$. For $f(x)=x^{2}(x - 3)(x + 3)$, the zero $x = 0$ has multiplicity 2, and the zeros $x=3$ and $x=-3$ have multiplicity 1. But we made an error above. The correct way is: since the multiplicity of $x = 0$ is 2, the graph touches the $x$ - axis at $x = 0$, and the graph crosses the $x$ - axis at $x=-3$ and $x = 3$.

Answer:

a) A. The graph of f(x) rises left and rises right.
b) $x=-3,0,3$
At which zeros does the graph of the function cross the x - axis? A. $x=-3,3$